Modelling the effects of social distancing, antiviral therapy, and booster shots on mitigating Omicron spread

As the COVID-19 situation changes because of emerging variants and updated vaccines, an elaborate mathematical model is essential in crafting proactive and effective control strategies. We propose a COVID-19 mathematical model considering variants, booster shots, waning, and antiviral drugs. We quantify the effects of social distancing in the Republic of Korea by estimating the reduction in transmission induced by government policies from February 26, 2021 to February 3, 2022. Simulations show that the next epidemic peak can be estimated by investigating the effects of waning immunity. This research emphasizes that booster vaccination should be administered right before the next epidemic wave, which follows the increasing waned population. Policymakers are recommended to monitor the waning population immunity using mathematical models or other predictive methods. Moreover, our simulations considering a new variant’s transmissibility, severity, and vaccine evasion suggest intervention measures that can reduce the severity of COVID-19.

shows the Bootstrap result with a thousand realization of estimated µ values. Each histogram indicates the distribution of estimated µ values from a thousand different datasets obtained by the Poisson error of the simulation result. Details of the statistical information from the distributions are in Table 1.
Bootstrapping gives us statistical information about the estimated parameter. Table 1 shows the mean, median, mode, standard deviation, 95% confidence interval, and estimated value from the data with the 1000 realization. The difference between the estimated value from the data and the mean value of all realization is lesser than 0.005, and standard deviations are lesser than 0.0005 for all µ. During the SD2, SD4, GR, and SGR, the ranges of µ(t) are (0.59,0.68), (0.64, 0.77), (0.49, 0.57), and (0.62, 0.66), respectively. Based on the estimation, the most eased phase is GR (µ = 0.54), and the most strict phase is SD4 (µ = 0.73).

B Model equations
Our model is an SEIQRD compartment model considering the severity of quarantined individuals. Each infection-related compartment (E, I, Q m , Q s , R, D) is categorized according to vaccination state and variant, which are distinguished by subscripts. The subscript j represents the type of variant. j = 1, 2, or 3 refers to infection with pre-Delta, Delta, or Omicron, respectively. The Subscript X represents the immune state of infected individuals. X = S; U ; or P refers to infected individuals from S, V ; U ; P 1 , P 2 , P 3 , W , V b , respectively. Equations 1 to 6 describe the infection-related states.
The susceptible compartments are also categorized according to vaccination state. S and V are unprotected groups since they are not vaccinated or need more time to get immunity. The ineffectively vaccinated group (U ) cannot get immunity by any vaccine. Any variant can infect the unprotected and ineffectively vaccinated groups. The effectiveness of vaccines against each variant is different, and hence, V may become P 1 , P 2 , or P 3 , which means fully protected against some or all variants. As vaccine-induced immunity of individuals in P j wane over time, they move to the W compartment. Individuals in W and U can get booster shots, but U cannot get immunity in our 3/6 assumption. Right after receiving booster shots, individuals in W move to V b , and eventually to P j .
The forces of infection consider the variants and their transmissibility, social distancing, and transmission reduction by vaccines. The forces of infection are defined as At the beginning of our estimation, vaccine, Delta, and Omicron-related compartments are zero because there are no vaccines and such variants. Since we have no data on infected people who are not yet confirmed, E S,1 (0) = 400/κ 1 = 1600 and I S,1 (0) = 400/α = 2400 are calculated from average daily confirmed cases at the end of February 2021. The initial values for Q s S,1 , Q m S,1 , R S,1 , and D S,1 are obtained from the data [1]. The initial value for S is calculated as the remaining number of individuals based on the total population and other initial values. With these, our proposed mathematical model consists of Equations 1-8 with initial values given in Table 2 Table 2. Initial value for model compartments; other initial of compartment are zero.

C Calculation of the parameters in our model
Vaccine-related model parameters (τ v j , e v j , e b j ) are calculated based on the effectiveness at 2 and 20 weeks from the second dose for mRNA and viral-vector type after inoculation of booster shot. Two weeks after primary vaccination, the effectiveness of Pfizer against Delta (Omicron) is 91% (65%) and of Moderna against Delta (Omicron) is 95% (75%). It reduced to 67% (11%) and 76% (15%) twenty weeks after primary vaccination. Since the proportion of the population in Korea who were given mRNA as primary doses is 75.4% and who got Moderna vaccine is 19.6%, we can calculate the total effectiveness of primary vaccine at 2 and 20 weeks from the second does. For example, the effectiveness of mRNA vaccine against Delta is 0.804 × 0.91 + 0.196 × 0.95 = 0.92, and the effectiveness of primary vaccine against Delta is 0.2467 × 0.83 + 0.7543 × 0.92 = 0.9 Similarly, weighted arithmetic mean is used to calculate e v j , e v j , e b j from e pre,v j , e pre,vw j , e pre,b j , respectively. The values of the parameters e pre j , e pre w,j , e pre b,j , which we found from the literature, are stated in Table 3. Also, we assume that vaccine-induced immunity wanes exponentially at a rate τ v j . Then h eff (t) := e v j e −τ v j t . Since we know two values of vaccine effectiveness at two different time points, then At last, f = 60.7% is computed as the percentage of dead over cumulative severe cases. All calculation results are shown in Table 3.  Table 3. The raw data for vaccine-related parameters. e v j , τ v j , e b j , and ω are used in the mathematical model.